The Symmetric-Product Origin of the Jacobian Counterexample:
Alpöge's Map is the Affine Insert Map on Sym³(ℙ¹),
with a Trace-Zero Fingerprint and a Dimension-Two Firewall

1  Setting and provenance

Alpöge's map is \(F=(F_1,F_2,F_3)\), \(F_1=u^3z+y^2u(4+3xy)\), \(F_2=y+3xu^2z+3xy^2(4+3xy)\), \(F_3=2x-3x^2y-x^3z\), \(u=1+xy\), with \(\det JF\equiv-2\) and the verified collision \(F(0,0,-\tfrac14)=F(1,-\tfrac32,\tfrac{13}{2})=F(-1,\tfrac32,\tfrac{13}{2})=(-\tfrac14,0,0)\) — a nonzero-constant Jacobian polynomial map that is not injective, falsifying the Jacobian Conjecture for \(n\ge3\) (announced July 20, 2026; Report I). Report II gave its structural anatomy (a torus action, an explicit \(3\!:\!1\) fold, a rigidity theorem); Report III built a two-parameter family of higher-weight lifts in dimension three. What was still missing — and what leading commentators explicitly named as missing — is a reason the object exists: a piece of classical geometry that the counterexample is a shadow of. This report supplies it. The geometric construction analysed here was proposed to us in outline (the map \(\pi\) on symmetric products, the tangent-not-osculating hyperplane, the affine complements); our contribution is to make it a coordinatised, machine-checked identity and to connect it rigorously to Alpöge's specific map and to the dimension-two obstruction. All equation-level claims are SymPy exact-arithmetic computations; the two structural inputs we borrow (Report II's Rigidity Theorem 6; the fact that units on \(\mathbb A^n\) are constants) are cited, not re-proved.

2  The insert map and why its Jacobian must be constant

Identify \(\mathrm{Sym}^2(\mathbb P^1)\cong\mathbb P^2\) (binary quadratics) and \(\mathrm{Sym}^3(\mathbb P^1)\cong\mathbb P^3\) (binary cubics). In the affine charts of monic polynomials, write the pair \(\{q,r\}\) as \(T^2-e_1T+e_2\) (\(e_1=q+r,\ e_2=qr\)) and the triple as \(T^3-\sigma_1T^2+\sigma_2T-\sigma_3\). Inserting the root \(p\) multiplies by \((T-p)\):

The forcing mechanism. Let \(X\) be the complement of \(R\) together with the preimage of a hyperplane \(H\subset\mathrm{Sym}^3\) chosen tangent-but-not-osculating to the small diagonal, and \(Y=\mathrm{Sym}^3\setminus H\); both are isomorphic to \(\mathbb A^3\), and \(\pi|_X:X\to Y\) is étale and generically \(3\!:\!1\). In affine coordinates \(\pi|_X\) is a polynomial self-map of \(\mathbb A^3\) whose Jacobian is a nowhere-vanishing regular function. But the only units of \(\mathbb C[x,y,z]\) are the nonzero constants. Hence the Jacobian is a nonzero constant, while the map remains genuinely three-to-one. The non-injectivity is not an accident to be discovered; it is the generic degree of a symmetric-product cover, and the constancy of the Jacobian is not a miracle but the algebra of units on affine space. This is the entire story in one sentence — the rest of the report makes "\(\cong\mathbb A^3\)" and "Alpöge's map" concrete and checked.

3  An explicit Keller specimen and the trace-zero fingerprint

Realising the affine chart concretely gives the explicit polynomial map \(\Phi_3:\mathbb A^3\to\mathbb A^3\) (the \(k=1,d=2\) member of the Report III construction, here re-interpreted as the Sym³ insert map):

Phi_1 = x^3 y^3 z + (3/2) x^2 y^4 + 3 x^2 y^2 z + (7/2) x y^3 + 3 x y z + 2 y^2 + z
Phi_2 = 3 x^3 y^2 z + (9/2) x^2 y^3 + 6 x^2 y z + 6 x y^2 + 3 x z + (1/2) y
Phi_3 = -x^3 z - (3/2) x^2 y + x

The generic degree three was independently confirmed numerically at several rational targets (the source point recovered plus two siblings), and — as a family check — the resultant eliminant of the \(k=1\) members has core degree exactly \(n=3,4,5\) in \(x\), matching "choose one of \(n\) roots."

4  A rational non-injectivity witness

5  Alpöge's map is this map, up to gauge

It remains to connect \(\Phi_3\) to Alpöge's specific \(F\). Both are torus-equivariant for the same one-parameter group.

This is the conceptual "why": the three collision points are the three root-splittings of one cubic; the constant Jacobian is the unit-group of \(\mathbb A^3\); the torus is the scaling that spreads the ramification into an orbit and renders the cover étale. The object is a classical symmetric-product cover wearing an affine disguise.

6  The dimension-two firewall

Why does the same machine not produce a plane counterexample — as it must not, since JC₂ is believed true and remains open? The construction that yields \(\Phi_3\) is governed by a one-variable master polynomial \(q_s(s)=\frac{k+1}{d-k}s^{k}-\frac{d+1}{d-k}s^{d}\) with \(1\le k

This dovetails with Report II's geometric reading — the counterexample lives over a \(\mathbb Z/2\) quotient singularity that the smooth plane \(\mathbb C^2/\langle\pm1\rangle\cong\mathbb C^2\) cannot supply — and with Report III's determinant law \(-k/(k+1)\), which likewise degenerates only at the excluded boundary. Three independent descriptions of the same wall.

7  Scope: what is and is not a symmetric-product map

Open frontier. The natural theorem this suggests — an explicit dimension-\(n\) insert-Keller map \(\Psi_n:\mathbb A^n\to\mathbb A^n\), generic degree \(n\), constant Jacobian, one uniform proof for all \(n\ge3\) — is not established here. Its content is precisely the explicit isomorphism \(X\cong\mathbb A^n\) (the tangent-not-osculating complement is affine space) for general \(n\), realised by polynomial coordinates. We have it for \(n=3\); the const-ifying coordinate change for \(n\ge4\) is the honest next target. If it exists it yields counterexamples in every dimension \(\ge3\) of the minimal possible generic degree; if it obstructs at some \(n\), that obstruction is itself a theorem about which dimensions the symmetric-product mechanism reaches.

8  Threats to validity

• Interpretation vs. computation. Theorems 2–5, 7 are exact CAS identities. The identification "Alpöge = Sym³ insert map" (Corollary 6) is a deduction from those computations plus Report II's Rigidity Theorem 6; if a subtle error in Report II's global gauge classification survived its independent audit, the identification would weaken to "same equivariance class, conjecturally one orbit." The direct evidence (same weights, same generic degree, same fold, explicit gauge siblings in Report II) is strong but is not a fresh closed-form gauge map from \(F\) to \(\Phi_3\); producing that explicit map is a worthwhile hardening step.
• The two \(\cong\mathbb A^3\) isomorphisms of §2 are asserted from the geometry (hyperplane complements and the tangent-not-osculating condition); our machine check bypasses them by exhibiting the explicit polynomial map \(\Phi_3\) and verifying its properties directly, so the paper does not depend on an unproven "complement is affine space" claim for \(n=3\). For general \(n\) that claim is exactly the open frontier of §7.
• Novelty is a human judgment. The symmetric-product picture is classical; framing Alpöge's map as its affine shadow, the trace-zero fingerprint, and the \((d-k)\) firewall are, to our knowledge, not previously recorded for this 4-day-old object — but a literature check is a task for expert reviewers.
• No Lean formalisation. Alpöge's original result was Lean-checked; this report is SymPy exact-arithmetic only, the same standard as Reports I–III. A Lean port of Theorems 2–4 and 7 is straightforward and not yet done.
• CAS trust base: SymPy exact rational/symbolic arithmetic; no floating point in any assertion; one command, seconds to replay.
• No expert human review yet. This draft exists to solicit it.

9  Artifacts

Replication (Python 3 + SymPy, no network, no floating point in assertions):
pip install -r requirements.txt && python verify_sym3.py → "status": "PASS".